Both main parties’ Brexit positions have been thrown into chaos days before a crucial set of Commons votes and with just two weeks until the UK is due to present a plan covering trade and the Irish border at a summit of EU leaders.

Prime Minister Theresa May’s Cabinet erupted into open warfare again as Brexiteer ministers led by David Davis rejected a draft “backstop” proposal to keep the Irish border open because it fails to set a date for the UK to finally leave the European Union customs regime. The plan was due to be presented by Mrs May’s top Europe adviser, Olly Robbins, to EU counterparts in Brussels tomorrow.

Meanwhile, Labour is facing its biggest ever internal rebellion under Jeremy Corbyn, with more than 100 MPs set to support the UK remaining in the single market via membership of the European Economic Area (EEA) against order from the party’s frontbench.

On Tuesday, 15 amendments to the EU Withdrawal Bill from government defeats in the House of Lords will be debated and voted on by MPs in a marathon 12-hour session. They include amendments seeking to keep the UK in the EU customs union and secure membership of the EEA.

In a bid to placate pro-EU MPs and party members, while not being seen to block Brexit, Mr Corbyn and shadow Brexit secretary Keir Starmer published an alternative amendment, supporting “full access to the internal market” without mentioning the EEA.

The plan has angered backbench MPs, who say the amendment is toothless and will fail, because Conservative rebels won’t back a measure put forward by the Labour leadership.

“It’s a ridiculous fudge,” one Labour MP told The Scotsman. “Right now no-one is interested in how Labour would negotiate Brexit, because we aren’t going to be in government before Tuesday. We have the chance to defeat the Tories now, and the frontbench have got to take it.”

Scottish Labour MEP Catherine Stihler called the move “deeply disappointing”, saying it “deliberately throws away a golden opportunity to save thousands of jobs in Scotland and across the UK”.

A group of Tory rebels has also sent a signal to Mrs May and Mr Corbyn, tabling an amendment to a different piece of legislation calling for the UK to join the EEA. The 13 Conservative MPs – a number big enough to defeat the government – include the Scottish Tory Paul Masterton.

At PMQs, SNP Westminster leader Ian Blackford added to pressure on both party leaders, asking: “While the leader of the Opposition is playing games, the question he should have asked today is, will the Prime Minister stop her charade and vote for the Lords amendments next week for membership of the EEA and the customs union, protecting jobs and prosperity?”

The Brexit sub-committee of the Cabinet will meet again today in a bid to sign off the four-page backstop proposal.

Downing Street sought to play down the row, saying that ministers had already “agreed the policy of what it would look like – and of course that there needs to be one”.

Mr Davis is also understood to want a White Paper setting out the UK’s post-Brexit trade and customs plans to be published ahead of an EU summit at the end of June.